Time-Series Forecasting

Ordinary least squares answers "what is the typical outcome here?" Quantile regression answers "what outcome should I expect to be exceeded only 10% of the time here?" It fits the same straight line, but penalizes a miss on one side more than a miss on the other: for the 90th percentile, undershooting costs nine times as much as overshooting, so the line is pushed up until exactly 90% of the points sit beneath it. Fit several tau values and you get a band, not a line, and the band can fan out where the data is noisier, which a single mean line cannot say.

Fit a shallow tree, look at what it got wrong, then fit another tree to those errors, and keep going. Each tree is deliberately weak — a few splits, shrunk by a small learning rate — so no single one can overfit, and the ensemble corrects itself gradually. Random Forest averages many independent strong trees to reduce variance; boosting chains many dependent weak trees to reduce bias, which is why the two behave so differently despite both being tree ensembles.

Three ideas stacked, and the acronym is in the wrong order for how they are applied. First integrate: difference the series until it is stationary, because everything that follows assumes the statistical behaviour does not drift. Then autoregress: predict the differenced value from its own recent values. Then add moving average: also predict it from the recent forecast ERRORS, which is how the model absorbs shocks that persist for a few periods without permanently changing the level. The AR part remembers where the series was; the MA part remembers where the model was wrong. That second memory is what distinguishes ARIMA from a regression on lag features, and it is why its intervals are calibrated rather than optimistic.

Forecast with a weighted average of the past where recent observations count for more, and the weights fall away geometrically. One parameter controls how fast the memory fades: near one and the model chases every wiggle, near zero and it barely moves. That is simple exponential smoothing, which handles a level. Add a second component that tracks how fast the level is changing and you have a trend. Add a third that tracks the repeating pattern and you have Holt-Winters. Each component is updated by the same rule — blend what you just observed with what you already believed — so the entire model is three numbers of state and three parameters, and it never looks at more than the last observation.

Separate what the system is doing from what you managed to measure. The state is the thing you care about — a position, a level, a trend — and it evolves by its own rules. The observation is a noisy, possibly incomplete look at it. Then run two steps forever. Predict: push the state forward and let the uncertainty grow, because time passing makes you less sure. Update: take the measurement and move the estimate toward it, by an amount that depends on which you trust more. That trade-off is the whole filter, and the Kalman gain is literally its arithmetic — when your prediction is uncertain and the sensor is good, the gain is near one and you believe the sensor; when the reverse, you barely move.

A recurrent cell that carries an explicit memory track alongside its hidden state, and learns three gates that decide what to erase from that memory, what to write into it, and what to expose. The memory passes forward through addition rather than repeated multiplication, which is the entire trick: gradients can travel back hundreds of steps without vanishing.

A convolution over time looks at a fixed window, which is too short to be useful and too expensive to widen by brute force. Two modifications fix that. Causal: pad only on the left, so a position never sees the future — without which the model cheats and the validation numbers are fiction. Dilated: skip a growing number of positions between taps at each layer, so a stack of layers with dilations 1, 2, 4, 8 reaches back sixteen steps with four layers rather than sixteen. The receptive field doubles per layer, so it grows exponentially in depth while the parameter count grows linearly. And because nothing is carried forward, every output position is computed from its own window independently — the entire sequence in one parallel pass.

A hundred traffic sensors are not a hundred independent series: congestion at one junction shows up downstream twenty minutes later, and the road network says which junctions are downstream. So alternate two operations — mix each node with its graph neighbours, then mix each node with its own recent past — and stack them. Spatial dependency comes from the graph rather than from being learned blind, which is the entire reason this beats a per-series model when the network is real.

Most forecasting inputs are not the same kind of thing. Store size never changes; a public holiday next Tuesday is known now; yesterday’s footfall was observed and tomorrow’s will not be. A plain sequence model that concatenates all of these into one vector per timestep either leaks future information or throws away covariates it legitimately has. TFT keeps the three kinds separate by construction, learns which variables matter at each step, and emits quantiles rather than a point — which is what a forecast is actually for, since the decision downstream is almost always asymmetric in cost.

Give a block the last few hundred observations and ask it for two things: a forecast, and a reconstruction of the input it thinks it explained. Subtract that reconstruction from the input and hand the remainder to the next block. Each block therefore works on what its predecessors could not explain, and the forecasts add up — which is gradient boosting’s idea rearranged into a neural network, and exactly the same idea as classical decomposition into trend and seasonality, except the components are learned instead of specified.

Instead of predicting tomorrow’s demand, predict the parameters of a distribution over tomorrow’s demand — a mean and a shape — and train by maximizing the likelihood of what actually happened. To forecast further than one step, sample a value from that distribution, feed it back as the next input, and continue; do that a few hundred times and you have a cloud of possible futures. Any quantile you need is then just a percentile of those trajectories, and crucially the uncertainty compounds correctly across the horizon because each step’s sample genuinely conditions the next.

Two ideas arriving at the same time. The first: a forecast and an imputation are the same operation — some values are known and some are not, and a conditional generative model fills in the unknown ones. Mask the future and you get forecasting; mask the middle and you get imputation; the model does not distinguish. The second: a model pretrained on millions of unrelated series learns what series look like in general, so it can forecast one it has never seen without being fitted to it. That is a genuinely new capability for forecasting, and it is also the one that most needs careful evaluation, because the corpus is enormous and the benchmarks are public.

Draw the straight line (or hyperplane) that comes closest to every point at once, where "closest" means the sum of squared vertical distances is as small as it can be. Every coefficient answers one question: if this feature goes up by one unit and nothing else moves, how much does the prediction change?

Ordinary least squares will happily hand two nearly identical features enormous coefficients of opposite sign, because only their sum is pinned down by the data. Put a price on coefficient size and that stops. An L2 price (Ridge) is smooth, so it shrinks correlated features toward each other and keeps all of them; an L1 price (Lasso) has a corner at zero, so it pushes most coefficients exactly to zero and hands back a subset. Elastic net charges both. In every case you are buying a large reduction in variance with a small amount of bias — deliberately fitting the training data worse in order to predict new data better.

Linear regression assumes what you are predicting is a real number with constant-variance noise. Counts are not: they cannot go negative, and a count with a mean of 100 is genuinely noisier than one with a mean of 2. A GLM keeps the linear predictor — a weighted sum of features — and changes two things around it. A link function maps that sum onto the range the response actually lives in, and a variance function states how the noise grows with the mean. Choose identity link and constant variance and you have recovered OLS; choose logit and mu(1-mu) and you have logistic regression; choose log and mu and you have Poisson regression. It is one algorithm with the distributional assumption written down instead of assumed away.

Instead of fitting one function, put a probability distribution over all the functions that could have produced the data, and let the observations narrow it. The kernel is what makes this concrete: it states how correlated the function values at two inputs should be, which is the same as stating how smooth, how wiggly, and how periodic you believe the function is. Conditioning that prior on the data gives a posterior that is again Gaussian, so the answer is closed-form — a mean and a variance at every input. The variance is the part that earns its keep: it is small where data is dense, and it widens automatically wherever you leave the data behind, which no point-estimate regressor will tell you.

A single linear layer can only draw a straight boundary. Put two of them back to back with a nonlinearity in between and the first layer can bend the space before the second one cuts it — which is enough to separate things no straight line could. Stack more and each layer works on features the previous one built rather than on the raw input. Training runs the network forward to see how wrong it is, then walks the error backwards through the layers, and at each step the chain rule says how much each weight contributed. Backpropagation is exactly that: the chain rule, with the shared intermediate results computed once instead of once per weight.

Slide a small learned filter across the input and record how strongly it responds at each position. Two assumptions are baked into that operation and they are the whole reason it works: nearby inputs are related (locality), and a feature worth detecting in one place is worth detecting everywhere (translation equivariance, via weight sharing). Stack these layers and the receptive field widens, so early layers see edges and later ones see objects.

A feedforward network sees a fixed-size input and has no way to represent "what came before". Give it a hidden state that is fed back in alongside the next input and it does: the state is a running summary of everything seen so far, updated once per step. Weights are shared across steps, so the same update rule applies at position three and position three hundred — which is what lets one model handle sequences of any length. The trouble is in the gradient. To learn from step three hundred what happened at step three, the error has to be multiplied back through three hundred copies of the same matrix, and repeated multiplication by a matrix either shrinks toward zero or grows without bound. Almost nothing in between.

A plain RNN overwrites its state at every step, which is why a signal from far back is gone by the time it would be useful. The fix is to let the network decide, per step and per dimension, how much to overwrite and how much to keep. That is the update gate: a number between zero and one that interpolates between the old state and a freshly computed candidate. Set it near zero and the state passes through untouched — and crucially, the gradient passes through with it, because carrying a value forward is addition rather than multiplication. The reset gate does the complementary job: it decides how much of the old state the candidate is even allowed to look at, which lets the unit forget deliberately rather than by decay.

Translating a sentence with a plain encoder-decoder means reading the whole source, compressing it into one fixed vector, and then generating from that vector alone. Everything the decoder will ever know about the source has to fit there — and it does not, which is why the approach degraded sharply on long sentences and why reversing the source words used to help. Attention removes the compression. Keep every encoder state, and at each output step compute a score between what the decoder is currently doing and each source position, turn those scores into weights that sum to one, and read a weighted average. The decoder is no longer remembering the source; it is looking at it, and choosing where to look.

Each node builds a representation by gathering messages from its neighbours, combining them, and updating itself — then repeats. After k rounds a node has absorbed information from everything within k hops. It is convolution generalized off the grid: a CNN assumes neighbours are the adjacent pixels, a GNN asks you to supply the adjacency explicitly. That single change is what lets it operate on molecules, payment networks, and road systems, where "nearby" is a relationship rather than a coordinate.

Every position looks at every other position and decides, from content alone, which ones matter to it. A convolution asserts in advance that neighbours are relevant; recurrence asserts that recency is. Attention asserts neither — it learns the routing. That is why it transfers across modalities, and also why it needs far more data than architectures whose prior is built in: it has to learn from scratch what the others assume.

A residual block computes h + f(h), which is Euler’s method for an ODE with a step size of one. Take that seriously: shrink the step, let the solver choose how many it needs, and the network stops being a fixed stack of layers and becomes a vector field that something else integrates. Two things follow that are not merely aesthetic. The trajectory is defined at every time, not just at layer boundaries, which is exactly what an irregularly sampled series needs. And because the dynamics are reversible, the backward pass can be recomputed instead of stored, so memory stops growing with depth.

Take the linear model, then squash its unbounded output through a sigmoid so it lands in [0, 1] and can be read as a probability. Each coefficient answers: how much does one unit of this feature move the log-odds? The decision boundary is still a hyperplane — the sigmoid changes what the output means, not what shapes the model can separate.

You have p candidate features and want a short list. Trying every subset means 2^p fits, so stepwise walks instead: start with nothing, add whichever single feature improves an information criterion the most, repeat until no addition helps. Backward elimination runs the same walk in reverse from the full model, and the bidirectional version lets a feature re-enter or leave at every step. Each step costs a handful of refits, which is why it dominated statistics software for decades. The catch is what the procedure hides: the data chose the model, and then the same data are used to report on it. The p-values, confidence intervals and R-squared printed for the final model are computed as though the features had been specified in advance, so they are optimistically wrong, and nothing in the output says so.

Do not build a model of the world. Keep every example you have seen, and when a new point arrives, find the handful that look most like it and copy their answer — a majority vote for a class, an average for a number. Everything hard about the method is hidden in "look most like it": the distance function is the model, and choosing it badly is a modelling error that no amount of tuning k will repair. The trade is stark and worth stating up front: training costs nothing, and inference costs everything.

Of all the hyperplanes that separate two classes, take the one that sits as far as possible from the nearest points on either side. That clearance is the margin, and maximizing it is a bet worth stating plainly: a boundary with room around it is more likely to survive new data than one that squeaks between training points. Two things follow. Only the points on or inside the margin — the support vectors — affect the answer, so most of the training set could be deleted without changing the model. And because the solution depends on the data only through inner products, replacing that inner product with a kernel produces a nonlinear boundary without ever computing the coordinates it lives in.

One Gaussian describes a single blob; real data is usually several, so describe it as a weighted sum of Gaussians and let the fit decide how much of each. The obstacle is that you never observe which component produced a point — that assignment is the latent variable — so you alternate. Given the current components, compute how responsible each is for each point. Given those responsibilities, refit each component to the points it owns, weighted by ownership. The defining feature is that ownership is fractional: a point between two clusters belongs 60/40 rather than being forced into one. k-means is this same loop with the responsibilities rounded to 0 and 1 and every covariance forced spherical and equal, which is exactly why k-means cannot represent an elongated or overlapping cluster and this can.

Ask a sequence of yes/no questions about one feature at a time, and after each answer ask the next question of whichever group you landed in. Each question is chosen greedily: try every feature and every threshold, and keep whichever split makes the two resulting groups most internally homogeneous. Recurse until the groups are small or pure, then predict the majority class or the mean of each. The result is a partition of the feature space into axis-aligned boxes with a constant prediction inside each — which explains both its strengths and its ceiling, since no amount of depth makes a box into a diagonal.

A single decision tree is accurate on average and wildly unstable — resample the data and you get a different tree. Averaging many trees cancels that instability, but only if the trees actually disagree, and trees grown on bootstrap samples of the same data do not disagree nearly enough: whichever feature is most predictive becomes the root split of almost every one of them. So the forest adds a second source of randomness that is easy to miss and does most of the work: at every split, only a random subset of the features may be considered. That forces trees to use different features, which decorrelates them, which is what makes the averaging pay.

Data with many correlated columns does not really occupy all those dimensions — it sits near a lower-dimensional plane inside them. PCA finds that plane: the direction along which the points spread out most, then the direction of most remaining spread perpendicular to it, and so on. Keep the first few and you have a compressed description that loses very little. The catch is in what "loses very little" means, and it is worth being suspicious of: PCA measures loss as squared distance, so it keeps whatever varies most. Variance is not the same as information, and the direction that separates two classes can easily be one of the ones you just discarded.

Summarize the data with k representative points. Start with k guesses, assign every observation to its nearest guess, then move each guess to the centre of what it just captured, and repeat until nothing moves. The objective it descends is the total squared distance from points to their own centre, and the two steps alternate between the two things that objective depends on — which assignment, and which centres. What makes it so widely used is that both steps are trivial and it converges quickly. What makes it so widely misused is that it always returns exactly k clusters, whether or not the data has any.

Stop thinking about distance to a centre and start thinking about connectivity. Build a graph where nearby points are joined, then look for a way to cut it into pieces that severs as few connections as possible. Two crescents that interlock have no separating hyperplane and no useful centroids, but they are two obviously distinct connected regions, and cutting the graph finds them. The trick that makes this computable is that the best cut, once relaxed from a hard yes-or-no assignment to a continuous one, is given by the smallest eigenvectors of the graph Laplacian — so a combinatorial problem becomes a linear-algebra one. Those eigenvectors are new coordinates in which the crescents have become blobs, and then k-means finishes the job.

Start with every point as its own cluster, find the two closest clusters, merge them, and repeat until one cluster remains. The record of those merges — which two joined, and at what distance — is the dendrogram, and cutting it at any height or at any K gives a flat clustering for free. Everything interesting hides in one word, closest: the distance between two clusters is not defined until you choose a linkage. Single linkage uses the nearest pair of points, so clusters join along thin bridges and chain into long streaks. Complete linkage uses the farthest pair, so clusters stay compact. Average linkage splits the difference. Ward merges whichever pair increases the within-cluster sum of squares least, which is k-means’ objective applied greedily, and it is the only one of the four that needs Euclidean geometry. The others need only a dissimilarity — no mean, no coordinates, no metric axioms — so this is the clustering you reach for when your data is word sets compared by Jaccard, embeddings compared by cosine, or time series compared by dynamic time warping.

Stand on any point and count how many others are within a fixed radius ε. If at least minPts are (the point itself included), you are standing somewhere dense, and the point is a core point. Two core points within ε of each other are in the same cluster, and so is anything that chains through a sequence of such steps — the cluster is the crowd you can walk across without ever crossing a gap wider than ε in a crowded place. Points within ε of a core point but not dense themselves are border points: they join the cluster, but you cannot walk onward through them. Everything left over is noise. Nothing in this asks for the number of clusters, and nothing assumes a shape — a ring, a crescent and a blob are all just connected regions of density — which is exactly where k-means, with its convex Voronoi cells, fails. What it asks for instead is a single, global notion of "close", which is the whole difficulty: ε is a distance, so it is only as meaningful as the metric that measures it, and one ε cannot describe two clusters of different density.

Imagine every data point is a voter and every data point is also a candidate. Each voter tells each candidate how much it would like that candidate to represent it, measured against its best alternative — that is the responsibility. Each candidate answers with how available it is, which depends on how much support it is already collecting from everyone else — that is the availability. The two messages are passed back and forth until the votes settle, and the points that end up voting for themselves are the exemplars. Nobody told the algorithm how many clusters to find; that falls out of how reluctant each point is to stand as a candidate, a number called the preference. The leaders are real data points, not averages, which means the only thing the method ever needs is a table of how similar each pair is — and that table can come from edit distance, from correlation, from anything that can be computed between two items.

A ratings matrix has millions of rows and columns and almost nothing in it — most users have touched almost nothing. The bet is that behaviour is driven by a small number of underlying factors, so the matrix is approximately low rank: describe each user by a short vector of tastes, each item by a short vector of attributes, and predict an interaction as the dot product between them. Nothing says what the factors mean, and they are not interpretable; what matters is that fitting them on the entries you observed lets you fill in the ones you did not. The crucial detail is that the sum runs only over observed entries — treating the blanks as zeros would be a completely different and much worse model.

Write the decision as a list of numbers, the cost as a weighted sum of them, and every rule as a weighted sum that must stay under a limit. Each rule cuts space in half with a flat wall, so the set of decisions that obey every rule is a many-faced solid, a polytope. A linear cost is a tilted plane, and sliding that plane as far as it will go while still touching the solid always ends on a corner. That one geometric fact is the whole algorithm: simplex starts at a corner and walks along edges to a neighbouring corner that is cheaper, and it can stop the moment no neighbour is cheaper, because a convex solid has no false peaks. Interior-point methods reach the same corner by a different route, cutting through the inside along a smooth path instead of along the skin. The second idea is what makes this explainable to a business: every wall has a price. The dual variable on the machine-hours constraint says what one more hour is worth, and it comes out of the same solve for free. Then comes the cliff. Require that some variables be whole numbers (build the plant or do not, ship whole pallets) and the feasible set stops being a solid and becomes a scatter of lattice points inside one. The problem becomes NP-hard, and the practical method is to solve the continuous version, branch on a fractional variable, and let the continuous solution serve as a bound that prunes most of the tree. Everything that follows is about how well the continuous relaxation approximates the integer problem.

Keep a crowd of candidate answers and let the better ones have more children. Score every candidate with the fitness function, pick parents with a bias toward the high scorers, cut two parents together to make a child that inherits pieces of each, flip a few bits at random so the crowd keeps trying things nobody had, and repeat. The whole method is one dial: how hard to lean on what already works (selection pressure, crossover) against how hard to keep looking (mutation, population size). Lean too hard and the crowd collapses into near-copies of one decent answer, which is premature convergence. Lean too softly and it is random search with extra steps. The reason to accept that cost is what the fitness function is allowed to be. It can be a simulator, a lab assay, a compiled binary, a human rating, a backtest. It can be discontinuous, noisy, or defined over schedules and subsets and graphs where there is no gradient to take. The price is sample efficiency: a gradient method on a differentiable objective will beat it by orders of magnitude, so the honest question is never "is a genetic algorithm good" but "have I run out of anything cheaper that applies".

Instead of asking a stack of layers to learn the mapping H(x) directly, ask it to learn only the correction F(x) = H(x) - x and add the input back: y = F(x) + x. If the best thing a block can do is nothing, it only has to push its weights toward zero, which is easy; a plain stack has to learn the identity map through several nonlinear layers, which turns out to be hard. That is the whole idea, and its consequence was surprising: before it, making a network deeper made it train worse, not just generalise worse. After it, hundreds of layers train with ordinary SGD, and the shortcut became the default way to wire any deep network.

Downsampling is how a network learns WHAT is in an image, and it is exactly what destroys WHERE it is. A U-Net refuses to choose. A contracting path (convolution, then downsample, repeated) trades resolution for context until the bottleneck sees the whole scene. An expanding path (upsample, then convolution) climbs back to input resolution. And at every scale the encoder feature map is CONCATENATED onto the decoder feature map, handing the decoder the fine spatial detail the pooling threw away, so it only has to decide what each pixel is rather than reconstruct where the boundary was. The output is a class distribution at every pixel, which is why it is called dense prediction. Concatenation is the point: ResNet adds its skip, which forces the two sources to share channels and be summed blindly, whereas a U-Net stacks them and lets the next convolution learn how much of each to trust.

Cut the image into a grid of 16-by-16 patches, flatten each one into a vector, and feed the resulting sequence to a plain text transformer with nothing changed. The model is told nothing about two-dimensional structure — not locality, not translation equivariance, not scale — beyond a learned position embedding per patch. That sounds like it should not work, and on a hundred thousand images it does not: below a certain data scale a convolutional network wins comfortably, because the biases a CNN hardcodes are true and free. Above that scale the transformer learns better biases than the ones we knew to build in, and overtakes.

A neural network is a differentiable function, so you can compute its derivatives exactly with the same machinery that computes its gradients. That means you can write down a differential equation, evaluate how badly the network violates it at a scatter of points, and make that violation the loss. The result is a mesh-free solver whose solution is a closed-form function you can differentiate anywhere — and, more usefully, a framework where unknown coefficients in the equation are just more parameters to fit alongside the network.

An autoencoder that is forced to be probabilistic: instead of mapping an input to a single point in latent space, the encoder outputs a small Gaussian, and the decoder must reconstruct from a sample of it. Because nearby latent points now decode to similar outputs, the latent space becomes continuous rather than a scatter of memorized codes — which is what makes sampling from it produce something new rather than noise.

Start from a simple distribution you can sample and evaluate — a standard Gaussian — and push it through an invertible function. The change-of-variables formula says exactly what happens to the density: it is the base density at the preimage, corrected by how much the transformation stretched or compressed space there. So if you can build a flexible invertible function whose local volume change you can compute cheaply, you get an exact likelihood for free. The entire field is a search for such functions, and every architecture in it is a different answer to the same question: how do you keep a Jacobian determinant tractable without making the map trivial?

Train a forger and a detective at the same time. The forger produces samples, the detective tries to tell them from real data, and each improves against the other. Neither ever sees an explicit description of what makes data realistic — the forger only ever learns from the detective’s mistakes. That is the appeal, because it means no likelihood needs to be defined and no reconstruction distance has to be chosen, which is why GAN images were sharp when everything else was blurry. It is also the problem: the target is moving, so there is nothing that decreases monotonically and no loss value that tells you how well it is going.

A plain GAN’s discriminator answers a classification question, and once it can answer it confidently it stops being informative — the gradient it hands back vanishes exactly when the generator most needs guidance. Replace it with a critic that scores rather than classifies, constrain how fast that score can change, and the answer becomes a distance that stays meaningful however far apart the two distributions are. Separately and independently: tell both networks what they are looking at. A generator given a class label learns one conditional distribution per label rather than one blurred mixture, and the critic can then check that the sample matches its label rather than merely looking real.

Generating an image in one step is hard; removing a little noise from a slightly noisy image is easy. So define a process that gradually destroys data into pure noise over a thousand steps, train one network to undo a single step of it, and then run that network a thousand times starting from noise. Nothing about any individual step is difficult, which is why this trains stably where a GAN does not — and the price is paid exactly where you would expect, at inference, where a thousand sequential network evaluations produce one sample.

Stop thinking about a thousand discrete denoising steps and think about a continuous path from noise to data. At any point along that path there is a direction that leads toward the data distribution — the score, or equivalently a velocity — and a network can be trained to predict it by regression. Once you have that direction field, generating a sample is solving a differential equation, which means every numerical-integration technique becomes available. Flow matching then asks the obvious follow-up: if the path is ours to choose, why not choose a straight one? A straight path can be traversed in very few large steps, which is the whole reason modern samplers need dozens rather than thousands.

Predict the next token, over and over, on everything ever written. The objective is trivial and the consequence is not: to predict the next token well you have to model syntax, then facts, then reasoning, then style, because all of those constrain what comes next. Nothing in the training says "learn arithmetic" — arithmetic is learned because it helps predict the token after an equals sign. That is the whole argument for scale, and it is also why the capabilities are uneven in ways that are hard to predict from the objective.

A causal model sees only the left context because it has to: the chain rule imposes an order. If you are not going to generate, that constraint buys nothing — a classifier reading a sentence has the whole sentence. So hide fifteen percent of the tokens, let every position attend in both directions, and predict what was hidden. The representations are strictly better for understanding tasks because each token is informed by what follows it as well as what precedes it. What you give up is the ability to generate and the ability to compute a likelihood, and both losses follow from the same fact.

A dense layer applies every parameter to every token, which ties capacity to cost: doubling the parameters doubles the arithmetic. Replace the layer with many copies and a small router that picks one or two per token, and the two quantities come apart — total parameters grow with the expert count while per-token compute stays fixed. The difficulty is immediate and is the whole subject: the router is trained by gradient descent on the primary loss, and the fastest way for it to reduce that loss early is to send everything to whichever expert is currently best, which starves the others of gradient and makes them permanently worse. So a separate auxiliary loss is added whose only job is to stop that, and tuning its weight against the primary loss is the real engineering.

Nothing here specifies what an embedding should be. It specifies only that paired things should end up closer than unpaired things, and the geometry falls out of enforcing that over millions of pairs. This is why the technique is defined almost entirely by its data: the encoder is ordinary, and the consequential choices are what counts as a positive pair, where the negatives come from, and the temperature that decides how hard the loss pushes on the closest wrong answer. Get the pairs right and a small encoder beats a much larger one. Get them wrong and the embeddings look fine on held-out pairs from the same distribution while failing on everything real, because the model learned to solve the sampling artefact rather than the task.

Fine-tuning a large model means changing every weight, which means storing a gradient and two optimizer moments for every weight — several times the model's own size in memory before a single activation is saved. The observation behind LoRA is that the CHANGE a fine-tune makes is far simpler than the model it changes: adapting a general model to legal drafting or to one company's support tone is not learning language again, it is nudging behaviour the model already has. So freeze the pretrained weight and learn a low-rank correction beside it, W0 + BA with an inner dimension of eight or sixteen instead of four thousand. Two consequences fall straight out. Training memory collapses, because optimizer state is proportional to trainable parameters and there are now a thousand times fewer. And the result is a small file that means nothing on its own, which is a feature — one frozen base can serve hundreds of tenants, each with their own adapter, if you are willing to keep it unmerged. The failure mode is the mirror image of the premise: when the task really does require capability the base does not have, a low-rank update cannot express it, and no amount of tuning reveals that the ceiling is structural rather than statistical.

Monte Carlo waits for the episode to finish and averages what happened. Temporal-difference learning refuses to wait: it takes one step, sees the reward, looks at its own estimate of where it landed, and updates toward that. The target is therefore partly made up — it contains the very quantity being learned — and the whole method is an argument that this is a good trade. It is. Substituting an estimate for the remaining trajectory removes almost all of the variance, because a single reward plus one value lookup is far less random than a sum over a hundred future steps, and the resulting method learns online, from incomplete episodes, in tasks that never terminate. What it buys with is bias and an assumption. The bias is real and does not vanish with data — a wrong value at the next state propagates backwards into this one. The assumption is that the state is Markov, because bootstrapping trusts the next state's estimate to summarize everything after it; where Monte Carlo is simply indifferent to state aliasing, this converges confidently to the wrong answer. Lambda is the dial between the two extremes, and the best setting is almost always in the middle.

Everything else in this category spends its effort on credit assignment across time. A bandit has no time to assign across — you pick an arm, you get a reward, and the next decision starts from the same place. What is left when the temporal structure is gone is the exploration-exploitation trade in its purest form: you can only learn about an arm by pulling it, and every pull spent learning is a pull not spent earning. Three answers matter. Epsilon-greedy explores at random with some probability, which is simple and wasteful, because it explores arms it already knows are bad just as often as promising ones. UCB is optimistic in the face of uncertainty: score each arm by its estimated mean plus a confidence bonus that shrinks as it is pulled, then take the best score. An arm that looks bad but has barely been tried keeps a high bonus and gets tried again; an arm that has been tried and is genuinely bad falls away. Thompson sampling is Bayesian and even simpler to state: keep a posterior over each arm's value, draw one sample from each, play the argmax of the samples. Arms the posterior is uncertain about sometimes draw high and get played, in exact proportion to the probability they are best. What makes this group worth studying is that it is the only one here with meaningful theory — regret bounds that hold, are tight, and are actually predictive of behaviour — and simultaneously the group that appears most often in production, usually in contextual form. It is also the group most often reached for wrongly, because the assumption it rests on, that actions do not change the state, fails quietly rather than loudly.

Every method so far has learned what to do without ever learning what happens. A transition gives a model-free agent one scalar of information — the reward — and it throws away the rest, which is the next state, a full observation of how the world responded. Model-based methods keep it. Fit a dynamics model that predicts the next state from the current state and action, and suddenly each interaction is a complete supervised example, which is why this branch routinely learns from one or two orders of magnitude fewer interactions than a model-free one. Once a model exists there are two things to do with it. Plan with it: search over action sequences by rolling them out in imagination, pick the best one, execute only its first action, then throw the rest away and replan from the new state. That last part sounds wasteful and is the entire mechanism — replanning every step means model error never gets a chance to accumulate, and it is why model predictive control is deployed across process industries where no other optimal-control method is. Or generate with it: use the model to synthesize experience and train an ordinary model-free agent on the mix, which is Dyna and its descendants. The catch is the one that keeps recurring in this category. A planner searching for the best action sequence under a learned model is an optimizer pointed at a learned function, and it will find the places where the model is wrong and optimistic — except here the errors compound step by step down the rollout, so a model that is slightly wrong per step can be arbitrarily wrong at horizon twenty. Every serious technique in this area is a response to that.

If an expert already solves the task, the obvious move is to record what they do and fit a model that maps states to their actions. That is behavioural cloning, it is ordinary supervised learning with no reward and no exploration, and it works far more often than the theory suggests it should — which is exactly why the way it fails is worth understanding precisely. The failure is not that the model is inaccurate. It is that the model is evaluated somewhere it was never trained. The demonstrations cover the states an expert visits, and an expert who is good rarely visits bad states, so there is almost no data about how to recover from a mistake. The moment the learner makes a small error it is slightly off the expert's distribution, where its predictions are slightly worse, which produces a slightly larger error, and the drift compounds. The cost of this is not linear in the episode length but quadratic, which is the formal statement of "a small imitation error becomes a large behavioural one". DAgger fixes it by changing what is labelled rather than how it is fitted: run the learner, collect the states IT visits, ask the expert what they would have done there, add those to the dataset, and repeat. Now the training distribution converges toward the one the policy is actually evaluated on, and the error becomes linear again. The catch is in the requirement — an expert who can be queried on arbitrary states, interactively, which is exactly what you usually do not have. Most of the field is about what to do when you only have a fixed set of demonstrations.