Lecture 6: Convexity and Training Difficulty

Convexity and Training Difficulty

Why does the shape of a loss decide whether training is easy?

Vishnu Boddeti

Example 1

Linear probes test what a frozen representation exposes

Frozen model activations

Fit a linear classifier

Predict an attribute

Concrete exampleA probe predicts sentiment from a language-model layer while the underlying model remains unchanged.

Convex probe training is predictable; end-to-end tuning changes both the representation and the decoder.

Understanding Intermediate Layers Using Linear Classifier Probes (Alain and Bengio, 2016)

Why is fitting a linear probe predictable while end-to-end transformer tuning is not?

A convex linear or logistic probe on frozen activations has no competing local minima.
End-to-end transformer tuning is non-convex and depends on initialization, optimizer state, and schedule.
What property of the loss landscape explains the difference?

Convex losses have no competing local minima

For a convex $f$, every chord lies on or above the graph: $f(\lambda x+(1-\lambda)y)\le\lambda f(x)+(1-\lambda)f(y)$.
The upward-facing bowl has no competing local minima: every local minimum is global.
A non-convex loss can have several basins, so different starting points may converge to different minima.

Choose two starting points and compare where they converge

Convex — $f(x) = a x^2 + b x$

Non-convex — $g(x) = a x^4 - b x^2$

Change convex curvature $a>0$: 1.0 Change the nonconvex well depth $b$: 1.0 Click each plot to choose a starting point, then watch gradient descent move to a minimum.

Click once on each landscape. The dotted trail shows every gradient-descent step.

Example 2

An adversarial threat model defines allowed changes

Clean input

Bounded perturbation set

Worst-case model output

Concrete exampleAn image may change by at most ε in every pixel while an attacker searches for a misclassification.

Robustness claims are meaningful only relative to a specified norm, radius, knowledge, and attack objective.

Explaining and Harnessing Adversarial Examples (Goodfellow, Shlens, and Szegedy, 2014)

How does an adversarial threat set define the size of a perturbation?

The threat set $K$ contains perturbations allowed at unit budget.
Scale $K$ until it first reaches $\delta$; that scale is $\gamma_K(\delta)=\inf\{t>0:\delta\in tK\}$.
If $K$ is convex, symmetric, bounded, and contains $0$ in its interior, this gauge is a norm.

A symmetric convex set defines a gauge

Let $K$ be convex, bounded, symmetric about the origin ($K=-K$), and contain $0$ in its interior.
Along the ray at angle $\theta$, compare $|x|$ with the boundary radius $R_K(\theta)$: $\|x\|_K=|x|/R_K(\theta)$.
Convexity gives the triangle inequality; symmetry gives $\|-x\|_K=\|x\|_K$.
These properties make $\|\cdot\|_K$ a norm whose unit ball is exactly $K$.

Reshape $K$, then measure a point with its gauge

Vertex radius $r_1$ (0°, 180°): 1.4 Vertex radius $r_2$ (60°, 240°): 1.0 Vertex radius $r_3$ (120°, 300°): 1.7 Drag the orange point and compare its distance with the boundary of $K$.

Interpreting $K$ as a threat set: gauge below 1 means the perturbation is allowed, 1 is the budget boundary, and above 1 is outside the threat model.

Example 3

Model merging interpolates learned solutions

Checkpoint A

Weighted parameter average

Merged checkpoint

Concrete exampleSeveral fine-tuned image classifiers are averaged to improve accuracy without additional training.

Merging works best when checkpoints occupy compatible regions of parameter space and share a common initialization.

Model Soups: Averaging Weights of Multiple Fine-tuned Models Improves Accuracy (Wortsman et al., 2022)

When can interpolating two model checkpoints produce a useful merged model?

checkpoint θ_A→interpolate→checkpoint θ_B

Model merging often starts with a line segment between parameter vectors.
Endpoints can each perform well, yet the interpolation may cross a loss barrier unless the checkpoints occupy a compatible basin or are first aligned.

Move along a checkpoint interpolation and expose a possible loss barrier

Move the control to compare the two normalized responses.

Interpolation coefficient α

Parameter averaging is easy; preserving capability along the path is the nonconvex part that must be measured.

Example 4

Safe policy mixtures need geometric guardrails

Two tool-use policies

Interpolate action probabilities

Check safety constraint

Concrete exampleAn agent blends a conservative policy with a faster experimental router only while the update remains inside a trust region.

Even plausible endpoints do not guarantee every mixture satisfies nonlinear safety or permission constraints.

Trust Region Policy Optimization (Schulman et al., 2015)

When can an agent blend tool policies without leaving a safe operating region?

safe routing policies→convex mixture weights→blended policy

If each candidate routing policy satisfies convex resource and risk constraints, a probability mixture remains feasible.
Nonconvex requirements—such as mutually exclusive permissions or discrete workflow invariants—do not inherit that guarantee.

Increase policy blending and compare capability coverage with boundary risk

Move the control to compare the two normalized responses.

Policy-mixture flexibility

Convex combinations preserve convex constraints, but agent workflows often contain discrete rules that must be checked separately.

Practice problems

Each scenario hides a mathematical structure from this lecture. Identify the structure, justify it, and work through the resulting problem.

Problem 1 Certified robustness

Translate a certificate between two threat models

A classifier is certified against perturbations of limited Euclidean length, but a deployment checklist measures only the largest change to any single feature. How conservative must the per-feature limit be before the old certificate still applies?

Model evidence

The input has dimension $d=64$, and the classifier is known to be safe whenever $\|\delta\|_2<0.24$.

What $\ell_\infty$ radius is sufficient to stay inside the certified $\ell_2$ ball?
Why must the conversion depend on the input dimension?

Problem 2 Distributional robustness

Does an uncertainty set define a valid notion of size?

A robust optimizer draws a diamond-shaped safe region, with changes to the first feature costing twice as much as changes to the second. It reports the amount by which the region must be enlarged to include a proposed perturbation. Is that report really a norm?

Model evidence

$C=\{\delta\in\mathbb{R}^2:2|\delta_1|+|\delta_2|\le1\}$ and $\gamma_C(x)=\inf\{t>0:x\in tC\}$.

What is $\gamma_C(x)$ in closed form?
Does it satisfy positivity, homogeneity, and the triangle inequality?
How does the unit ball encode the unequal cost of changing the two features?

Problem 3 Function approximation

When are two learned functions close enough?

A student model's predictions look increasingly like its teacher's on $[0,1]$. Yet their input sensitivities keep oscillating. Whether the models count as “close” depends on what the evaluator chooses to measure.

Model evidence

$f_n(x)=x+\frac{1}{n}\sin(nx)$ and the teacher is $f(x)=x$.

Do the predictions converge uniformly to the teacher's predictions?
Do their derivatives converge uniformly as well?
What conclusions do you obtain with the ordinary sup norm and with $\|g\|=\|g\|_\infty+\|g'\|_\infty$?