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Batch size counts the examples combined in a mini-batch. In the simple stochastic gradient descent loop where each group produces one parameter update, it controls how many examples contribute to that update. It differs from a training epoch, which counts a pass over the available training data.^[ref]

For an original counting example, partition ten examples with nominal batch size $b=4$ and retain the remainder. The actual sizes are $4,4,2$, giving $\lceil 10/4\rceil =3$ updates per training epoch. Discarding the remainder instead gives two updates and uses only eight examples. More generally the retained-remainder update count is $\lceil n/b\rceil$ for $n$ examples, provided there is one update per group.

Batch size also changes gradient sampling noise. If independently sampled component gradients at a fixed parameter have scalar variance ${\sigma }^2$, their sample mean hasFor an original exact example, suppose one component gradient is equally likely to be 0 or 2. Its mean is 1 and variance is 1. A two-example mini-batch gradient takes values $0,1,2$ with probabilities $1/4,1/2,1/4$, giving variance $1/2$. This calculation holds parameters fixed and assumes independent sampling; sampling without replacement has a finite population correction.

With gradient accumulation, several groups may contribute before an update. Two full four-example groups then involve eight examples per update if their gradients are normalized consistently. State whether a reported size counts the processed group or all examples contributing to an update. Lower sampling variance alone does not establish faster training for a fixed computation budget.

A cosine learning rate schedule varies the learning rate along a segment of a cosine curve. For a single decreasing segment, declare $t$ to count completed schedule advances and choose a duration $T>0$. With no competing rate updates, setIt starts at ${\eta }_{\max }$, reaches the midpoint of the two bounds at $T/2$, and ends at ${\eta }_{\min }$. This is the cosine segment used in SGDR; repeating such segments with restarts is an additional policy, not part of the single-segment formula.^[ref]

For an original example, let ${\eta }_{\max }=0.1$, ${\eta }_{\min }=0.01$ and $T=4$. At $t=0,1,2,3,4$ the rates are approximately $0.1,0.08682,0.055,0.02318,0.01$. Differentiating the continuous interpolation givesThus the rate decreases throughout the interior when the upper bound exceeds the lower bound, and its interpolation has zero slope at both ends. A smooth rate curve does not by itself prove convergence of the learning procedure.

The clock must be declared: advancing once per training epoch differs from advancing after every parameter update. The PyTorch CosineAnnealingLR documentation explicitly distinguishes its implementation without restarts from SGDR. Its example advances the scheduler after the training and validation operations. To stop at the lower bound, stop or hold the schedule at the declared endpoint; indefinitely extending the cosine formula is not a permanently decreasing rule.^[ref]

A learning rate schedule specifies how the learning rate changes during an iterative learning procedure. Its rule can depend on the update count, a training epoch count or another declared event. The rule and the clock must both be stated: changing a rate every parameter update is different from changing it after a complete pass through the training data.

A simple step-decay example iswhere $t$ counts completed schedule advances from zero. The values at $t=0,1,2$ are $0.1$, those at $t=3,4,5$ are $0.01$, and the value at $t=6$ is $0.001$. The StepLR implementation documents periodic multiplication by a specified factor; its example advances the scheduler after the training and validation operations.^[ref]

A different schedule admits an exact convergence calculation. For gradient descent on $L(\theta )={\theta }^2/2$, take ${\eta }_t=1/(t+2)$. With no other update terms,Multiplying the successive ratios gives a telescoping product:Starting at ${\theta }_0=4$ gives $4,2,4/3,1,\dots$. This proves convergence for the stated deterministic quadratic, rather than for every objective or noisy learning procedure.

The underlying stochastic gradient descent update also depends on its gradient estimate and any additional terms, not just the scheduled scalar.^[ref] Report the initial index, update order and any external changes to the rate. The StepLR documentation explicitly permits other changes alongside its own, so the isolated step formula above assumes that no competing rule intervenes.^[ref]

A mini-batch is a group of training examples processed together in one iteration of a learning algorithm. Its batch size counts examples, while a training epoch usually counts one pass over the available training data. In stochastic gradient descent, a mini-batch gradient combines the component gradients from the selected examples; the group of examples and the resulting numerical estimate are distinct objects.^[ref]

For an original counting example, partition eight examples into mini-batches with nominal batch size three and retain the final partial group. The sizes are $3,3,2$, giving three updates in one training epoch if each group produces one update. Discarding the final partial group changes which observations participate.

Unequal group sizes also matter when reporting a loss function. At fixed model parameters, suppose the three mini-batch mean losses are $1,1,10$. Their unweighted mean is $4$, but the mean over all eight examples isThe smaller last group must not receive the same weight as a larger group when the intended statistic is a per-example sample mean. This bookkeeping issue is separate from the randomness used to select a mini-batch.

SGDR means stochastic gradient descent with warm restarts. Its learning rate schedule repeatedly lowers the learning rate along a cosine curve, then raises it to begin another cycle while retaining the current parameters. Ilya Loshchilov and Frank Hutter's method uses this warm restart to continue an existing optimization trajectory.^[ref]

In cycle $i$, let $s$ be elapsed training time since the last restart and $T_i>0$ the cycle duration, measured in the same units. The cosine learning rate schedule isIt starts at ${\eta }_{\max ,i}$, reaches the midpoint of the two bounds at $s=T_i/2$, and reaches ${\eta }_{\min ,i}$ at the cycle endpoint. The original implementation updates the elapsed training epoch count within each mini-batch, allowing fractional epoch progress.^[ref]

For an original schedule, choose bounds 0 and 0.1, initial duration $T_0=4$ epochs and duration multiplier 2. The successive cycle lengths are 4, 8 and 16 epochs, with restart boundaries at total elapsed times 4, 12 and 28. In the first continuous cycle, the rates at $s=0,2,4$ are respectively $0.1,0.05,0$.

A discrete convention must specify which update uses the endpoint. For example, sampling phases $s=0,1,2,3$ in the first cycle and restarting before the next step gives rates approximately $0.1,0.08536,0.05,0.01464$, followed by $0.1$. The zero endpoint in the continuous curve is then a limit, rather than an additional zero-rate optimizer step.

Retaining parameters also has a concrete meaning. In an original deterministic gradient descent example with $f(\theta )={\theta }^2/2$ and current $\theta =0.5$, resetting only the schedule leaves $\theta =0.5$. The next step at learning rate0.1 gives0.45; a step at0.01 would give0.495. The restart changes the next movement rather than drawing a new initial parameter. Handling a momentum buffer is a further state convention; restarting the schedule and discarding all optimizer state are different operations. Record cycle phase, durations and rate bounds with an optimization checkpoint.

A training epoch is conventionally one pass through the training data used by an iterative learning procedure. Several parameter updates can occur within that pass. A mini-batch supplies a group of examples for an update, while the epoch counts progress through the data collection rather than counting individual updates.^[ref]

Suppose there are $N=10$ training examples. Visit each exactly once per pass, form mini-batches of at most $B=4$ examples and retain the smaller final group. Their sizes are $4,4,2$. With one update for every mini-batch, an epoch hasupdates. Five epochs therefore make fifteen updates and fifty example visits. Fifty visits do not mean fifty distinct examples: the same ten examples are reused.

Changing the mini-batch size changes the update count. Taking all ten examples together gives one update per epoch under the same rule, while groups of two give five. An epoch count alone consequently does not specify the number of stochastic gradient descent steps, the amount of computation or the final fitted model.

The finite-pass assumption matters. Dropping an incomplete final group in this example gives two updates and only eight visits during that iterator traversal. Reordering the examples on later passes can change which examples are omitted. Sampling with replacement can repeat some examples before visiting others, so a chosen number of sampled updates need not be a literal once-per-example pass. Report those conventions rather than assuming that every loop named epoch guarantees full coverage.

The learning rate schedule needs a declared clock too: changing the learning rate once per epoch differs from changing it once per mini-batch. The optimization tutorial separates the outer loop from the operations that clear gradients, perform backpropagation and update parameters inside the training loop.^[ref] Keeping data passes and update steps distinct makes training runs easier to compare.